Μεταθετική Άλγεβρα
Μεταθετική Άλγεβρα associative algebra thumb|300px| [[Άλγεβρα ---- Στοιχειώδης Άλγεβρα Αφηρημένη Άλγεβρα Γραμμική Άλγεβρα Μεταθετική Άλγεβρα Υπολογιστική Άλγεβρα Ομολογιακή Άλγεβρα Παγκόσμια Άλγεβρα Αλγεβρική Αριθμοθεωρία Αλγεβρική Γεωμετρία Αλγεβρική Συνδυαστική ]] thumb|300px|Άλγερα [[Μαθηματική Μήτρα|μητρών.]] thumb|300px| [[Μαθηματικά Γεωμετρία Άλγεβρα Μαθηματική Λογική Μαθηματική Ανάλυση Διακριτά Μαθηματικά Τοπολογία Γραμμική Άλγεβρα Στατιστική Οικονομικά Μαθηματικά ]] - Ένας Επιστημονικός Κλάδος των Μαθηματικών. Ετυμολογία Η ονομασία "μεταθετική" σχετίζεται ετυμολογικά με την λέξη "μετάθεση". Εισαγωγή In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A'' the structure of a ring; the addition and scalar multiplication operations together give ''A the structure of a vector space over K''. In this article we will also use the term K-algebra to mean an associative algebra over the field ''K. A standard first example of a K''-algebra is a ring of square matrices over a field ''K, with the usual matrix multiplication. In this article associative algebras are assumed to have a multiplicative unit, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R'', instead of a field: An 'R''-algebra' is an ''R''-module with an associative R''-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if ''S is any ring with center C'', then ''S is an associative C''-algebra. Definition Let ''R be a fixed commutative ring (so R'' could be a field). An '''associative ''R-algebra''' (or more simply, an R-algebra) is an additive abelian group A'' which has the structure of both a ring and an [[module |''R-module]] in such a way that the scalar multiplication satisfies : r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all r'' ∈ ''R and x'', ''y ∈ A''. Furthermore, ''A is assumed to be unital, which is to say it contains an element 1 such that : 1 x = x = x 1 for all x'' ∈ ''A. Note that such an element 1 must be unique. In other words, A'' is an ''R-module together with (1) an ''R''-bilinear map A'' × ''A → A'', called the multiplication, and (2) the multiplicative identity, such that the multiplication is associative: : x(yz) = (xy)z\, for all ''x, y'', and ''z in A''. (Technical note: the multiplicative identity is a datum,Put in another way, there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras. while associativity is a property. By the uniqueness of the multiplicative identity, "unitarity" is often treated like a property.) If one drops the requirement for the associativity, then one obtains a non-associative algebra. If ''A itself is commutative (as a ring) then it is called a commutative R-algebra. Υποσημειώσεις Εσωτερική Αρθρογραφία * Μεταθετική Άλγεβρα * Μη-Μεταθετική Άλγεβρα * Γραμμική Άλγεβρα * Τοπική Άλγεβρα * Γεωμετρία Βιβλιογραφία * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia Κατηγορία:Επιστημονικοί Κλάδοι Μαθηματικών